All elements of a finite-dimensional unital associative algebra are algebraic

Let $𝐴$ be a [[K-monoid| $𝕂$ -ring]]. If $𝐴$ is finite dimensional then every element in $𝐴$ is algebraic, since the series ${𝑎 𝑖} \infty 𝑖 = 1$ is linearly dependent. falg